Model order reduction with oblique projections for large scale wave propagation.

*(English)*Zbl 1329.86018Summary: There are many applications in which it is necessary to solve large scale parametrized wave propagation problems repeatedly. This is still quite a challenging task, even with the largest available computer clusters. In this paper we will discuss the application of Model Order Reduction (MOR) to problems in seismic petroleum exploration, with the aim of diminishing the necessary computing time by a significant factor. We consider POD and some variants. POD is a Model Order Reduction technique that uses snapshots of a few simulations in order to quickly compute related problems with similar accuracy. The method of lines via a Petrov-Galerkin approximation that uses the snapshots as basis functions is the considered approach. The order reduction comes from projecting the wave equation discretized in space to the subspace spanned by the snapshots. This has been shown earlier to work well in two dimensions. The challenge in three dimensions comes from the size of the spatial meshes required and the fact that the method usually requires a number of snapshots that do not fit in fast memory, even for current high end multicore machines. Parallelization is not an option since it is already used for other aspects of this massive problem.

##### MSC:

86A15 | Seismology (including tsunami modeling), earthquakes |

76Q05 | Hydro- and aero-acoustics |

74L05 | Geophysical solid mechanics |

PDF
BibTeX
XML
Cite

\textit{V. Pereyra}, J. Comput. Appl. Math. 295, 103--114 (2016; Zbl 1329.86018)

Full Text:
DOI

##### References:

[1] | Kaelin, B.; Pereyra, V., Fast wave propagation by model order reduction, Electron. Trans. Numer. Anal., 30, 406-419, (2008) · Zbl 1171.65072 |

[2] | Pereyra, V., Wave equation simulation in two-dimensions using a compressed modeler, Am. J. Comput. Math., 3, 231-241, (2013) |

[3] | Chunling Wu, Dimitri Bevc, V. Pereyra, Model order reduction for efficient seismic modeling, in: SEG Annual Meeting, 2013. |

[4] | Keller, H. B.; Pereyra, V., Symbolic generation of finite difference formulae, Math. Comp., 32, 955-971, (1978) · Zbl 0387.65049 |

[5] | Halko, N.; Martinsson, P. G.; Tropp, J. A., Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53, 217-288, (2011) · Zbl 1269.65043 |

[6] | Mahoney, M. W., Randomized algorithms for matrices and data, Found. Trends Mach. Learn., 3, 123-224, (2011), NOW Publishers · Zbl 1232.68173 |

[7] | M.W. Mahoney, Approximate computation and implicit regularization for very large-scale data analysis, in: Proc. of the 2012 ACM Symposium on Principles of Database Systems, 2012, pp. 143-154. |

[8] | Hansen, P. C.; Pereyra, V.; Scherer, G., Least squares data Fitting with applications, (2013), Johns Hopkins University Press Baltimore · Zbl 1270.65008 |

[9] | Meng, Xiangrui; Saunders, Michael A.; Mahoney, Michael W., LSRN: a parallel iterative solver for strongly over- or under-determined systems, SIAM J. Sci. Comput., 36, C95-C118, (2014) · Zbl 1298.65053 |

[10] | Pereyra, V.; Rosen, J. B., Computation of the pseudoinverse of a matrix of unknown rank. Stanford university computer sciences report SC18, (1964) |

[11] | Rosen, J. B., Minimum and basic solutions to singular linear systems, SIAM J., 12, 152-162, (1964) · Zbl 0156.16205 |

[12] | Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient nonlinear model leduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Internat. J. Numer. Methods Engrg., 86, 1-25, (2011) |

[13] | Drineas, P.; Kannan, R.; Mahoney, M. W., Fast Monte Carlo algorithms for matrices I: approximating matrix multiplication, SIAM J. Comput., 36, 132-157, (2006) · Zbl 1111.68147 |

[14] | K. Schwarz, Darts, dice, and coins: sampling from a discrete distribution. Manuscript, 2011. |

[15] | Vose, M. D., A linear algorithm for generating random numbers with a given distribution, IEEE Trans. Softw. Eng., 17, 972-974, (1991) |

[16] | P.-G. Martinsson, V. Rokhlin, Y. Shkolnisky, M. Tygert, ID: a software package for low-rank approximation of matrices via interpolative decompositions. Version 0.4. Yale University, New Haven, Conn., 2014. |

[17] | Tropp, Joel A., Column subset selection, matrix factorization, and eigenvalue optimization, (SODA’09 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, (2009), SIAM Philadelphia, PA), 978-986 |

[18] | Krebs, J. R.; Anderson, J. E.; Hinkley, D.; Neelamani, R.; Lee, S.; Baumstein, A.; Lacasse, M.-D., Fast full-wavefield inversion using encoded sources, Geophysics, 74, WCC177-WCC188, (2009) |

[19] | Ikelle, L., Coding and decoding: seismic data modeling, acquisition and processing, SEG Ext. Abstr., 77, 66-70, (2007) |

[20] | Romero, L. A.; Ghiglia, D. C.; Ober, C. C.; Morton, S. A., Phase encoding of shot records in prestack migration, Geophysics, 65, 426-436, (2000) |

[21] | Neelamani, R.; Krhon, C. E.; Krebs, J. R.; Deffenbaugh, M.; Anderson, J. E.; Romberg, J. K., Efficient seismic forward modeling using simultaneous random sources and sparsity, SEG Expand. Abstr., 78, 2107-2111, (2008) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.